- #1

- 256

- 45

## Main Question or Discussion Point

I have begun teaching myself Lagrangian field theory in preparation for taking the plunge into quantum field theory ( it's just a hobby, not any kind of formal course ). When working through exercises, I have run across the following issue which I don't quite understand. I am being given a Lagrangian density, and ask to derive the equations of motion; I understand the principles involved, and everything is fine and easy until I get to the point where I need to evaluate the following expression :

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2[/tex]

wherein ##\varphi(x,y,z,t)## is a scalar field. My approach to this was as simple as it was naive :

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2=\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )[/tex]

which evaluates to ##\partial _{\mu}\varphi## via the product rule. However, this is where I get stuck, because the answer is wrong - the correct approach should have been

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )[/tex]

which apparently evaluates to ##\partial ^{\mu}\varphi## ( though I have difficulties with that as well, but that's a separate issue ), and leads to the correct equations of motion. My question is : why is ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )## and not ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )## ? I know that this is probably something very elementary, so please don't laugh at me, but I genuinely don't get it.

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2[/tex]

wherein ##\varphi(x,y,z,t)## is a scalar field. My approach to this was as simple as it was naive :

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )^2=\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )[/tex]

which evaluates to ##\partial _{\mu}\varphi## via the product rule. However, this is where I get stuck, because the answer is wrong - the correct approach should have been

[tex]\frac{1}{2}\frac{\partial }{\partial \left ( \partial _{\mu}\varphi \right )}\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )[/tex]

which apparently evaluates to ##\partial ^{\mu}\varphi## ( though I have difficulties with that as well, but that's a separate issue ), and leads to the correct equations of motion. My question is : why is ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial ^{\mu}\varphi \right )## and not ##\left ( \partial _{\mu}\varphi \right )^2=\left ( \partial _{\mu}\varphi \right )\left ( \partial _{\mu}\varphi \right )## ? I know that this is probably something very elementary, so please don't laugh at me, but I genuinely don't get it.